2. WHIM HEATING AND ION TEMPERATURE EVOLUTION

The plasma ion heating in the WHIM is most likely due to
cosmological shocks. The Alfvén Mach number of a shock propagating
through an ionised gas of local overdensity
=
/
< > at
the epoch z in the standard
CDM cosmology is
determined by

(1)

where B-9 is the magnetic field just before the shock,
measured in nG and vs7 is the shock velocity in
107cm s-1, and
< > is
the average density in the Universe.

The sonic Mach number for a shock propagating in a plasma of
standard cosmic abundance is

(2)

where T4 is the plasma ion temperature measured in
104 K (typical for a preshock photoionised plasma) and
fei = Te / Ti. An
important plasma parameter is

It is the ratio of the thermal and
magnetic pressures. In hot X-ray clusters of galaxies the beta
parameter is ~ 100 for ~ µG magnetic fields in the
clusters. The most uncertain parameter is the magnetic field value
in the WHIM allowing for both
~ 1 and
>>
1 cases.

In a supercritical collisionless shock the conversion of kinetic
energy of an initially cold flow to the ion distribution with high
kinetic temperature occurs in the thin ion viscous jump. The width
of the ion viscous jump
vi in a
collisionless shock propagating through a plasma with
~ 1 is
typically of the order of a ten to a hundred times of the ion inertial
length li defined as li =
c/pi 2.3 ×
107n-0.5 cm. Here
pi is the ion
plasma frequency. The ion inertial length in the WHIM can be estimated as

The width of the shock
transition region for magnetic field is also
10
li for a quasi-perpendicular shock, but it is often
about ten times wider for quasi-parallel shocks.

Properties of nonrelativistic shocks in a hot, low magnetised
plasma with high
>>
1 are yet poorly studied. Measurements from the ISEE 1 and
ISEE 2 spacecrafts were used by
Farris et
al. (1992)
to examine the terrestrial bow shock
under high beta conditions. These measurements were compared with
and found to be in agreement with the predicted values of the
Rankine-Hugoniot relations using the simple adiabatic approximation
and a ratio of specific heats,
, of
5/3. Large magnetic field
and density fluctuations were observed, but average downstream
plasma conditions well away from the shock were relatively steady,
near the predicted Rankine-Hugoniot values. The magnetic
disturbances persisted well downstream and a hot, dense ion beam was
detected leaking from the downstream region of the shock. The
observation proved the existence of collisionless shocks in high
beta plasma, but a detailed study of high beta shock structure is
needed for cosmological plasmas.

We discuss in the next section the ion heating in collisionless
shocks illustrating the most important features of the process
with the results of a hybrid simulation of the oxygen ions heating
in a quasi-perpendicular shock considered earlier by
Bykov et al. 2008
- Chapter 7, this volume.

Ion heating mechanisms in collisionless shocks depend on the
shock Alfvén Mach number, the magnetic field inclination angle
(n), plasma
parameter
and the composition of the incoming plasma flow. The structure of a
supercritical shock is governed by the ion flows instabilities (see e.g.
Kennel et
al. 1985,
Lembege et
al. 2004,
Burgess et
al. 2005).
In a quasi-parallel shock
(n 45°) a mixed
effect of a sizeable backstreaming ion fraction and the ions
scattered by the strong magnetic field fluctuations (filling the
wide shock transition region) results in the heating of ions in the
downstream region. The ions reflected and slowed down by an electric
potential jump at the shock
ramp of a quasi-perpendicular
(n 45°) shock
constitute a multi-stream distribution just behind a relatively thin
magnetic ramp as it is seen in Fig. 1 and
Fig. 2 (left panel). The O VII phase
densities and distribution functions were simulated with a hybrid
code for a quasi-perpendicular
(n =
80°) shock in a hydrogen-helium dominated plasma (see
Bykov et al. 2008
- Chapter 7, this
volume). Phase densities x - vx,
x - vy, x - vz of
the O VII ion are shown in Fig. 1 in the
reference frame
where the particle reflecting wall (at far right) is at rest and the
shock is moving. The shock is propagating along the x -axis from
the left to the right and the magnetic field is in the x-z
plane. The system is periodic in the y dimension. The incoming
plasma beam in the simulation was composed of protons (90%),
alpha particles (9.9%) and a dynamically insignificant fraction of
oxygen ions (O VII) with the upstream plasma parameter
~ 1. The ions do not change their initial charge states in a few
gyro-periods while crossing the cosmological shock ramp where the
Coulomb interactions are negligible.

Figure 1. O VII phase density in a hybrid
simulated quasi-perpendicular shock (80° inclination). The shock is
moving from left to right in the reference frame where the
particle reflecting wall is at rest. The figures show the oxygen
phase densities in x - vx, x -
vy and x - vz
projections from top to bottom respectively. The size of the
simulation box in x-dimension is about 300 li.

Figure 2. Hybrid simulated O VII
distribution function (normalised) as a function of a random velocity
component vy = vy -
<vy> transverse to the downstream
magnetic field in a quasi-perpendicular shock (80°
inclination). The shock propagates along the x-axis, while the
initial regular magnetic field is in the x-z plane. In
the left panel the distribution in the viscous velocity jump is shown. The
right panel shows the distribution behind the jump at the position
of the left end in Fig. 1. Multi-velocity
structure of the flow is clearly seen in the left panel, while it is
relaxing to quasi-Maxwellian in the right panel.

The simulated data in Fig. 1 show the ion velocities
phase mixing resulting in a thermal-like broad ion distribution at
a distance of some hundreds of ion inertial lengths in the shock
downstream (see the right panel in Fig. 2). It
is also clear in Fig. 1 that the shocked ion
distribution tends to have anisotropy of the effective temperature. The
temperature anisotropy
T
~ 3 T|| relative to the magnetic field was found in
that simulation. Moreover, the hybrid simulation shows that the
T of
the O VII is about 25 times higher than the effective
perpendicular temperature of the protons. Thus the ion downstream
temperature declines from the linear dependence on the ion mass. The
simulations show excessive heating of heavy ions in comparison with
protons.

Lee & Wu (2000)
proposed a simplified analytical model to estimate
the ion perpendicular temperature dependence on Z / A,
where mi = Amp. Specifically, the model predicts the ratio of the
ion gyration velocity vig2 in the downstream of a
perpendicular shock
(n ~
90°) to the velocity of the incident ion in the shock upstream,
v1,

where = 2 e /
mpv12 < 1,
and the potential jump is calculated
in the shock normal frame (see
Lee & Wu 2000).
The model is valid for the ions
with gyroradii larger than the shock transition width
vi.
It is not a fair approximation for the protons, but it is much better for
temperature estimation of heavy ions just behind the shock magnetic
ramp. The model of ion heating in the fast, supercritical
quasi-perpendicular
(n 45°) shocks of
a 3 predicts a
higher downstream perpendicular
temperature for the ions with larger A / Z.

The initial electron temperature just behind the viscous ion jump of
a cosmological shock depends on the collisionless heating of the
electrons. The only direct measurements of the electron heating by
collisionless shocks are those in the Heliosphere. The
interplanetary shock data compiled by
Schwartz et
al. (1988)
show a modest, though systematic departure of the electron heating from
that which would result from the approximately constant ratio of the
perpendicular temperature to the magnetic field strength (i.e.
adiabatic heating). Thus, some modest non-adiabatic electron
collisionless heating is likely present. In the case of a
nonradiative supernova shock propagating through partially
ionised interstellar medium the ratio Te /
Ti in a thin layer
(typically < 1017 cm) just behind a shock can be tested using
the optical diagnostics of broad and narrow Balmer lines (e.g.
Raymond 2001).
High resolution Hubble Space Telescope (HST) Supernova remant
(SNR) images make that approach rather attractive. A simple scaling
Te / Tivsh-2 was suggested by
Ghavamian et
al. (2007)
to be consistent with the optical observations of SNRs.

Strong shocks are thought to transfer a sizeable fraction of the bulk
kinetic energy of the flow into large amplitude nonlinear waves in
the magnetic ramp region. The thermal electron velocities in the
ambient medium are higher than the shock speed if the shock Mach
number s
< (mp / me)1/2,
allowing for
a nearly-isotropic angular distribution of the electrons. Non-resonant
interactions of these electrons with large-amplitude turbulent
fluctuations in the shock transition region could result in
collisionless heating and pre-acceleration of the electrons
(Bykov & Uvarov
1999,
Bykov 2005).
They calculated the electron
energy spectrum in the vicinity of the shock waves and showed that
the heating and pre-acceleration of the electrons occur on a scale
of the order of several hundred ion inertial lengths in the vicinity
of the viscous discontinuity. Although the electron distribution
function is in a significantly non-equilibrium state near the shock
front, its low energy part can be approximated by a Maxwellian
distribution. The effective electron temperature just behind the
front, obtained in this manner, increases with the shock wave
velocity as Tevshb with b 2. They
also showed that if the electron transport in the shock transition
region is due to turbulent advection by strong vortex fluctuations
of the scale of about the ion inertial length, then the nonresonant
electron heating is rather slow (i.e. b 0.5). The highly
developed vortex-type turbulence is expected to be present in the transition
regions of very strong shocks. That would imply that the initial
Te / Tivsh(b-a) just behind the
transition region would decrease with the shock velocity for
s >>
1. Here the index a is defined by the relation
Tivsha for a strong shock. The degree of
electron-ion equilibration in a collisionless shock is a declining
function of shock speed. In the case of strong vortex-type
turbulence in the shock transition region one expects in the
standard ion heating case with a = 2 and rather small b to
have (a - b)
2. That
Te / Ti scaling is somewhat
flatter, but roughly consistent with, that advocated by
Ghavamian et
al. (2007).
On the other hand in a collisionless shock of a moderate strength
s < 10 the
electron transport through the magnetic ramp region could be diffusive,
rather than by the turbulent advection by a strong vortexes. That
results in a larger degree of the collisionless electron
heating/equilibration in the shocks as it is shown in Fig. 4 of the
paper by
Bykov & Uvarov
(1999).
Recently,
Markevitch
& Vikhlinin (2007)
argued for the collisionless heating/equilibration of the electron
temperature in the bow shock of
s ~ 3 in the
1E 0657-56 cluster.

If the local Mach number
s of the
incoming flow in a strong shock wave exceeds
(mp / me)1/2, which
could occur in the cluster accretion shocks, the thermal electron
distribution becomes highly anisotropic and high frequency whistler
type mode generation effects could become important.
Levinson (1996)
performed a detailed study of resonant electron
acceleration by the whistler mode for fast MHD shock waves. Electron
heating and Coulomb relaxation in the strong accretion shocks in
clusters of galaxies was discussed in details by
Fox & Loeb (1997).

We summarise this section concluding that a collisionless shock
produces in the downstream flow a highly non-equilibrium plasma
state with strongly different temperatures of the electrons and
ions of different species. Moreover, the postshock ion
temperatures are anisotropic. The width of the collisionless shock
transition region is smaller by many orders of magnitude than the
Coulomb mean free path (that is of a kiloparsec range). We
consider now the structure and the processes in the postshock
Coulomb equilibration layers in the WHIM.